ELEMENTS OF GEOMETRY. INTRODUCTION. DEFINITIONS OF TERMS. 1. QUANTITY is any thing which can be increased, diminished, and measured. To measure a thing, is to find out how many times it contains some other thing, of the same kind, taken as a standard. The assumed standard is called the unit of measure. 2. In GEOMETRY, there are four species of quantity, viz.: LINES, SURFACES, VOLUMES, and ANGLES. These are called GEOMETRICAL MAGNITUDES. Since the unit of measure of a quantity is of the same kind as the quantity measured, there are four kinds of units. of measure, viz.: Units of Length, Units of Surface, Units of Volume, and Units of Angular Measure. 3. GEOMETRY is that branch of Mathematics which treats of the properties, relations, and measurement of the Geometrical Magnitudes. 4. In Geometry, the quantities considered are generally represented by means of the straight line and curve. The operations to be performed upon the quantities, and the rela tions between them, are indicated by signs, as in Analysis. The following are the principal signs employed: The Sign of Addition, +, called plus: Thus, AB, indicates that B is to be added to A. The Sign of Subtraction, called minus : Thus, AB, indicates that B is to be subtracted from A. The Sign of Multiplication, x: Thus, A x B, by B. indicates that A is to be multiplied A3, Thus, As, indicates that A is to be taken three times. as a factor, or raised to the third power. The Radical Sign, v Thus, VA, B, indicate that the square root of A, and the cube root of B, are to be taken. When a compound quantity is to be operated upon as a single quantity, its parts are connected by a vinculum or by a parenthesis: Thus, A + B x C, indicates that the sum of A and B is to be .. cates that the sum of A and B is to be divided by C. indi A number written before a quantity, shows how many times it is to be taken. Thus, 3 (A + B), indicates that the sum of A and B is to be taken three times. Thus, ABC, indicates that A is equal to the sum of B and C. The expression, A B+ C, is called an equation. The part on the left of the sign of equality is called the first member; that on the right, the second member. The Sign of Inequality, <: Thus, VA VB, indicates that the square root of A is less than the cube root of B. The opening of the sign is towards the greater quantity. The sign, .. is used as an abbreviation of the word hence, or consequently. The symbols, 1°, 2°, etc., mean 1st, 2d, etc. 5. The general truths of Geometry are deduced by a course of logical reasoning, the premises being definitions and principles previously established. The course of reasoning employed in establishing any truth or principle is called a demonstration. 6. A THEOREM is a truth requiring demonstration. 7. An AXIOM is a self-evident truth. 8. A PROBLEM is a question requiring solution. 9. A POSTULATE is a self-evident Problem. Theorems, Axioms, Problems, and Postulates, are all called. Propositions. 10. A LEMMA is an auxiliary proposition. 11. A COROLLARY is an obvious consequence of one or more propositions. 12. A SCHOLIUM is a remark made upon one or more propositions, with reference to their connection, their use, their extent, or their limitation. 13. An HYPOTHESIS is a supposition made, either in the statement of a proposition, or in the course of a demonstration. 14. Magnitudes are equal to each other, when each contains the same unit an equal number of times. 15. Magnitudes are equal in all respects, when they may be so placed as to coincide throughout their whole extent; they are equal in all their parts when each part of one is equal to the corresponding part of the other, when taken either in the same or in the reverse order. ELEMENTS OF GEOMETRY. BOOK I. ELEMENTARY PRINCIPLES. DEFINITIONS. 1. GEOMETRY is that branch of Mathematics which treats of the properties, relations, and measurements of the Geometrical Magnitudes. 2. A POINT is that which has position, but not magnitude. 3. A LINE is that which has length, but neither breadth nor thickness. Lines are divided into two classes, straight and curved. 4. A STRAIGHT LINE is one which does not change its direction at any point. 5. A CURVED LINE is one which changes its direction at every point. When the sense is obvious, to avoid repetition, the word line, alone, is commonly used for straight line; and the word curve, alone, for curved line. 6. A line made up of straight lines, not lying in the same direction, is called a broken line. 7. A SURFACE is that which has length and breadth without thickness. |